Proceedings of the Twenty-Third International Joint Conference on Artificial Intelligence
Towards a Deeper Understanding
of Nonmonotonic Reasoning with Degrees
Marjon Blondeel∗
Advisors:
†
Steven Schockaert , Dirk Vermeir ‡ , Martine De Cock §
1
Introduction
Answer set programming (ASP) [Baral, 2003] is a form of
declarative programming that can be used to model combinatorial search problems in a concise and declarative manner. One
of the advantages of ASP is the fact that it works nonmonotonically. On the contrary, classical logic works monotonically:
when new knowledge is added, the set of conclusions that
can be inferred grows. Hence it is not really suitable to imitate human reasoning since humans constantly revise their
knowledge when new information becomes available. In ASP,
nonmonotonicity is obtained by using a special operator “not”,
the negation-as-failure operator. An expression of the form
not a is “true” when we fail to derive a, whereas ¬a, the
negation of a, is “true” if we can derive ¬a.
For example, consider the following ASP program.
r1 : suitable
r2 : certificate
this problem. One solution to this problem is to allow propositions to be true to a certain degree in [0, 1] and to generalize
the syntax and semantics of ASP using fuzzy logics [Hájek,
1998]. We can then write a rule
← certificate ∧ not criminal-record
← true
Rule r1 informally means that a person is suitable for a job if
there is no reason to think that he has a criminal record and
if we can establish that he has a certificate. A rule such as
r2 is called a fact; the head “certificate” is unconditionally
true. Given such a program, the idea is to find a minimal set of
literals that can be derived from the program. These “answer
sets” then correspond to the solutions of the original search
problem.
Unfortunately, ASP is not directly suitable for expressing
continuous optimization problems since it is limited to expressing problems in Boolean logic. For example, suppose
one wants to travel by car from one city to another in winter.
The driving time that is needed to do this depends on several
factors; for instance the amount of snow, the distance and
the traffic. These concepts are a matter of degree rather than
Boolean properties, thus we cannot directly use ASP to model
∗
Vrije Universiteit Brussel, Department of Computer Science,
Pleinlaan 2, 1050 Brussel, Belgium, mblondee@vub.ac.be, Funded
by a joint Research Foundation-Flanders (FWO) project
†
Cardiff University, School of Computer Science and Informatics,
5 The Parade, Cardiff, CF24 3AA, s.schockaert@cs.cardiff.ac.uk
‡
Vrije Universiteit Brussel, Department of Computer Science,
Pleinlaan 2, 1050 Brussel, Belgium, dvermeir@vub.ac.be
§
Ghent University, Department of Applied Mathematics, Computer Science and Statistics (S9), Krijgslaan 281, 9000 Gent, Belgium,
martine.decock@ugent.be
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driving time ← f (snow, distance, traffic)
where “driving time”, “snow” , “distance” and “traffic” now
have to be seen as atoms that can be assigned a degree in [0, 1].
The function f defines how these degrees have to be combined
to establish the driving time.
Fuzzy answer set programming (FASP) (see [Blondeel et
al., 2013c] for an introduction to the topic) is a generalization
of answer set programming (ASP) based on fuzzy logics. In
FASP, a continuous search problem is translated into a set of
rules α ← β where the body β and the head α are built from
literals, expressions of the form not a with a a literal, constants
and connectives that can in principal be interpreted by arbitrary
[0, 1]n → [0, 1]-mappings. Such a rule now intuitively means
that the truth degree of α must be greater than or equal to the
truth degree of β. Answer sets of FASP programs are then
mappings from the set of literals into [0, 1].
Since it is a relatively new concept, little is known about
the computational complexity of FASP and almost no techniques are available to compute answer sets of FASP programs.
Furthermore, the connections of FASP to other paradigms of
nonmonotonic reasoning with continuous values are largely
unexplored. In our dissertation, we contribute to the ongoing
research on FASP on two different levels:
• Complexity issues (Section 2) We have pinned down
the complexity of the direct syntactical generalization
of classical ASP to FASP, and we have developed an
implementation into bilevel linear programming for this
type of programs. For another class of FASP programs
with more syntactic freedom, we showed a connection to
an existing open problem about the complexity of integer
equations, indicating that settling the complexity of FASP
programs in this case will not be an easy task.
• Connection to fuzzy modal logics (Section 3) We combined the paradigms of fuzzy logic and autoepistemic
logic into fuzzy autoepistemic logic, and showed that the
latter generalizes FASP. Furthermore, we have introduced
generalizations of classical propositional modal logics of
belief, and we have obtained a generalization of a known
result on the relationship between stable expansions, belief sets and “only believing” operators. Our ongoing
work deals with fully developing this “only believing”
logic to obtain a better understanding of the nature of
fuzzy autoepistemic logic and hence of FASP.
2
Complexity issues
We have analyzed the computational complexity of FASP
under Łukasiewicz semantics in [Blondeel et al., 2013b].
Łukasiewicz logic is a particular kind of fuzzy logic that is
often used in applications because it preserves many desirable
properties from classical logic. Given a FASP program P , a
literal l and a value λl ∈ [0, 1] ∩ Q, we are interested in the
following decision problems:
1. Existence: Does there exist an answer set I of P ?
2. Set-membership: Does there exist an answer set I of P
such that I(l) ≥ λl ?
3. Set-entailment: Is I(l) ≥ λl for each answer set I of
P?
For programs with rules of the form
a1 ⊕ . . . ⊕ an ← b1 ⊗ . . . ⊗ bm ⊗ not c1 ⊗ . . . ⊗ not ck
where ⊕ and ⊗ are resp. the Łukasiewicz disjunction and conjunction, we showed that the complexity of the main reasoning
tasks is located at the first level of the polynomial hierarchy although for classical ASP it is located at the second level. This
is due to the fact that we can use linear programming to verify
yes instances of the decision problems in polynomial time.
Moreover, we provided a reduction from reasoning with such
FASP programs to bilevel linear programming, thus opening
the door to practical applications.
For the subclass of programs in which negation-as-failure is
not allowed and where there is at most one literal in the head
of each rule, we showed that the answer set can be found in
polynomial time. Surprisingly, when allowing disjunctions to
occur in the body of rules – a syntactic generalization which
does not affect the expressivity of ASP in the classical case
– the picture changes drastically. Reasoning tasks are then
located at the second level of the polynomial hierarchy and
for simple FASP programs, i.e. programs in which negationas-failure and negation do not occur and in which each rule
has exactly one atom in the head, we have not been able to
pin down the complexity yet. More specifically, we have an
algorithm to find the unique answer set of these programs
in pseudo polynomial time but whether this problem is NPcomplete remains an open problem. Moreover, the connection
to an existing open problem about the complexity of integer
equations [Bjorklund et al., 2003] suggests that the problem
of fully characterizing the complexity of FASP in this more
general setting is not likely to have an easy solution. Another
possibly even more intriguing and fundamental open research
question is whether there exists a FASP program in which each
head contains at most one literal without answer sets. Other
future work is to study the complexity under other semantics.
3
Connection to fuzzy modal logics
Autoepistemic logic is an important formalism for nonmonotonic reasoning. It extends propositional logic by offering
the ability to reason about an agent’s (lack of) beliefs. More
precisely, these beliefs are a set of sentences in a propositional
language augmented by a modal operator B. Moreover, autoepistemic logic is well known to generalize the stable model
semantics of ASP [Gelfond and Lifschitz, 1988]. In [Blondeel
et al., 2013a] we combined the ideas of autoepistemic logic
and fuzzy logic to a fuzzy autoepistemic logic which can be
used to reason about one’s beliefs in the degrees to which
properties are satisfied. We showed that, like in the classical
case, fuzzy autoepistemic logic generalizes FASP.
On the other hand, there are well known links between
autoepistemic logic and several nonmonotonic modal logic
systems. In ongoing work we are investigating how fuzzy
autoepistemic logic can be studied in many-valued modal
settings. In particular, we have introduced generalizations of
classical propositional modal logics of belief based on finitelyvalued Łukasiewicz logic for which we obtained completeness
w.r.t appropriate Kripke style semantics and for which we
could prove NP-completeness for the satisfiability problem.
We obtained a generalization of Levesque’s [Levesque, 1990]
result on the relationship between stable expansions, belief
sets and “only believing” operators. In future work, by fully
developing this ”only believing” logic we want to obtain a
better understanding of the nature of fuzzy autoepistemic logic
and hence of FASP.
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